Hydrophilic side-chain polymer, electrolyte membranes

ABSTRACT

Exemplary methods for the characterization of proton dissociation and transport for hydrophilic components of hydrated Polymer Electrolyte Membranes (PEM&#39;s) is described. Disclosed features and specifications may be variously implemented, controlled, adapted or otherwise optionally modified to improve differential hydrophilicity of the sidechain of any ionomeric PEM material. A representative embodiment of the present invention generally provides for the amelioration of electro-osmotic drag of water by protons, for example, in Direct Methanol Fuel Cells.

FIELD OF INVENTION

The present invention generally concerns fuel cell technology; and moreparticularly, in one representative and exemplary embodiment, thecharacterization of proton dissociation and transport for hydrophiliccomponents of Polymer Electrolyte Membranes (PEM's). A PEM materialcomprising a novel differential sidechain chemical composition is alsodisclosed as inter alia preventing or otherwise amelioratingelectro-osmotic drag as the membrane material experiences increasedswelling upon permeation with, for example, aqueous methanol.

BACKGROUND

Fuel cells are electrochemical cells in which a free energy changeresulting from a fuel oxidation is converted into electrical energy. Theearliest fuel cells were first constructed by William Grove in 1829 withlater development efforts resuming in the late 1930's with the work ofF. T. Bacon. In early experiments, hydrogen and oxygen gas were bubbledinto compartments containing water that were connected by a barrierthrough which an aqueous electrolyte was permitted to pass. Whencomposite graphite/platinum electrodes were submerged into eachcompartment and the electrodes were conductively coupled, a completecircuit was formed and redox reactions took place in the cell: hydrogengas was oxidized to form protons at the anode (e.g., “hydrogenelectrode”) and electrons were liberated to flow to the cathode (e.g.,“oxygen electrode”) where they subsequently combined with oxygen.

Since that time, interest in the development of viable commercial andconsumer-level fuel cell technology has been renewed. In addition tovarious other benefits compared with conventional methods, fuel cellsgenerally promise improved power production with higher energydensities. For example, a typical hydrogen-oxygen cell operating atabout 250° C. and a pressure of about 50 atmospheres yieldsapproximately 1 volt of electric potential with the generation of waterand a small quantity of thermal energy as byproducts. More recently,however, modern Polymer Electrolyte Membrane Fuel Cells (PEMFC's)operating at much lower temperatures and pressures (i.e., on the orderof about 80° C. and about 1.3 atmospheres) have been observed to producenearly the same voltage potential.

An additional advantage of fuel cells is that they generally have ahigher energy density and are intrinsically more efficient than methodsinvolving indirect energy conversion. In fact, fuel cell efficiencieshave been typically measured at nearly twice those of thermo-electricconversion methods (i.e., fossil fuel combustion heat exchange).

With respect to portable power supply applications, fuel cells functionunder different principles as compared with standard batteries. As astandard battery operates, various chemical components of the electrodesare depleted over time. In a fuel cell, however, as long as fuel andoxidant are continuously supplied, the cell's electrode material is notconsumed and therefore will not run down or require recharging orreplacement.

One class of fuel cells currently under development for general consumeruse are hydrogen fuel cells, wherein hydrogen-rich compounds (i.e.,methanol) are used to fuel the redox reaction. As chemical fuel speciesare oxidized at the anode, electrons are liberated to flow through theexternal circuit. The remaining positively-charged ions (i.e., protons)then move through the electrolyte toward the cathode where they aresubsequently reduced. The free electrons combine with, for example,protons and oxygen to produce water—an environmentally clean byproduct.

Current interest in perfluorinated ionomers, such as, for example,NAFION® (a perfluorinated polymer available from DuPont MicrocircuitMaterials, E. I. Du Pont de Nemours and Company, 14 T.W. AlexanderDrive, Research Triangle Park, N.C., USA), stems from their potentialuse as polymer electrolyte membranes in fuel cell applications. See, forexample, R. Lemons, J. Power Sources 29, 251 (1990). NAFION is a phaseseparated material with a crystalline region consisting of a hydrophobicTEFLON® (also available from DuPont Microcircuit Materials, E. I. DuPont de Nemours and Company, 14 T.W. Alexander Drive, Research TrianglePark, N.C., USA) backbone and a hydrophilic ionic domain comprisingrandomly attached long pendant chains terminating with sulfonic acidgroups. The terminal acid functionality is generally analogous to thatof trifluoromethane sulfonic acid (e.g., triflic acid).

NAFION belongs to a class of polymers referred to as ion-containingpolymers (e.g., ‘ionomers’). Although extensive work has been undertakento characterize ionomers, the state and structure of ion aggregation andthe resulting modifications occurring upon hydration have not been wellunderstood, even though the importance of such considerations hasgenerally been appreciated. See, for example, A. Eisenberg, H. L. Yeager(Eds.), Perfluorinated lonomer Membranes, ACS Symp. Ser. 180, AmericanChemical Society, Washington, D.C. (1982).

Early small-angle x-ray scattering and thermo-rheological studiessuggest that the ions in NAFION are clustered, containing somefluoro-carbon material. See, for example, S. C. Yeo, A. Eisenberg, J.Appl. Polymer Sci. 21, 1875 (1977). Ion clustering was further supportedby both wide- and small-angle diffraction studies on hydrolyzed NAFION.See, for example, T. D. Gierke at al., J. Polymer Sci. Polymer Physics,Ed. 19, 1687 (1981); W. Y. Hsu, T. D. Gierke, Macromolecules 15, 101(1982); and W. Y. Hsu, T. D. Gierke, J. Memb. Sci. 13, 307 (1983). Thiswork provided a microstructure model where the structure of the systemwas proposed as consisting of an inverted micelle with —SO₃ groupsforming hydrated clusters embedded in the fluorocarbon phase withdiameters from 40 to 50 Å. It was further concluded from infraredstudies of water (H₂O, D₂O and HDO) in NAFION that the hydrated ionclusters were either much smaller than earlier estimated or were highlynon-spherical in shape with frequent local intrusions of thefluorocarbon phase. See, for example, M. Falk, Can. J. Chem. 58, 1495(1980). This work seemed to indicate that a substantial proportion ofwater molecules were exposed to the fluorocarbon environment. Morerecently, however, through reexamination of the data, a lamellarmorphology was proposed for NAFION that, upon hydrolysis, creates polarsulfonic acid domains of relatively large surface area parallel to oneanother and connected by tie molecules. See, for example, M. H. Lift,Polymer Preprints 38, 80 (1997). A similar micelle structure wasexperimentally observed for NAFION solubilized in DMF. See, for example,A. V. Rebrov at al., Polymer Science U.S.S.R. 32, 251 (1990).

As NAFION membranes may function both as separators and electrolytes infuel cell applications, the overall performance of the fuel cell isstrongly influenced by the conductivity of the membrane, which itself isa function of the state of hydration of the membrane. See, for example,T. A. Zawodzinski et al., J. Electro-Chem. Soc. 95, 6040 (1991); T. E.Springer et al., J. Electro-Chem. Soc. 138, 2334 (1991); T. A.Zawodzinski et al., J. Electrochem. Soc. 140, 1981 (1993); and T. A.Zawodzinski et al., Solid State Ionics 60, 1993 (1993). The watercontent of the membrane may largely be determined inter alia by theinterplay of at least three processes: (a) water absorption by themembrane; (b) transport of water through the hydrated membrane by meansof, for example, the protonic current (e.g., electro-osmotic drag); and(c) water diffusion effected by means of, for example, water activitygradients. Experimental measurements of electro-osmotic drag for varioussulfonated membranes over a wide range of water content have suggestedthat wall effects tend to dominate proton transport and that themechanism responsible involves the tethering of sulfonic acid groupsbound to water. Until recently, however, no molecular-levelunderstanding was available for electro-osmosis. See, for example, S. J.Paddison, T. A. Zawodzinski, Solid State Ionics 113-115, 333-340 (1998).

In Direct Methanol Fuel Cells (DMFC's), aqueous methanol (CH₃OH) isintroduced at the anode where the fuel is electrochemically oxidized toproduce CO₂, protons and electrons. With conventional catalysts(typically carbon supported platinum or platinum alloys) and under thecurrent operating temperature limitations (on the order of about 110°C.) not all of the methanol is oxidized. Due to the miscible nature ofwater and methanol, along with the permeability of the PolymerElectrolyte Membrane (PEM), the latter is adsorbed by the membranecausing inter alia increased swelling (essentially due to an increase inthe water concentration) of the membrane. During operation of the fuelcell, protonic current within the membrane drags water (e.g.,‘electro-osmotic drag’) from the anode to the cathode, which reduces theefficiency of the fuel cell by hindering the reduction reaction at thecathode (e.g., ‘cathode flooding’). This, in turn, generally requiresthe utilization of relatively expensive water management techniquestypically involving capture or return of water to the anode side of theDMFC. The electro-osmotic drag coefficient (e.g., the number of watermolecules dragged per proton) typically increases substantially uponswelling of the membrane. Accordingly, despite the efforts of the priorart, one problem warranting resolution is the characterization of theincreased electro-osmotic drag of water by protons in membranes used inthe manufacture of PEMFC's. Accordingly, in one representative andexemplary aspect, the present invention proposes a molecular-basednon-equilibrium statistical mechanical method for predicting diffusioncoefficients for sulfonic acid based PEM's. Moreover, a representativenovel ionomeric PEM material is described for preventing or otherwiseameliorating electro-osmotic drag.

SUMMARY OF THE INVENTION

In various representative aspects, the present invention provides asystem and method for characterizing and subsequently customizingionomeric membranes that generally tend not to experience substantialincreases in electro-osmotic drag upon exposure to, for example, aqueousmethanol. In an exemplary application, a novel ionomer is disclosed asinter alia exhibiting increased hydrophilicity upon swelling—generallywithout any substantial penalty in acidity and thus, consequent protonconductivity—such that increased protrusion of the sidechain may occurin the membrane water-filled pores with the terminal anionic groupsgenerally preserving at least some “structuring” of the water whilesubstantially preventing or otherwise ameliorating any increase inelectro-osmosis.

The disclosed method for modeling of the dielectric saturation inPolymer Electrolyte Membranes (PEM's) demonstrates that with increasedprotrusion of the anionic groups within the membrane pores, thepermittivity (i.e., dielectric constant) of the water generallydecreases. The decrease in the dielectric constant (relative to bulkwater) generally corresponds to the water being more constrained. Boththe predictive values and experimental measurements have shown thatunder such conditions, electro-osmosis is minimized.

In application to NAFION and PEEKK membranes, at various levels ofhydration, the present invention demonstrates improved predictivecapability—without requiring the use of ‘fitting parameters’ and/orinput from SAXS experiments or molecular orbital calculations. Diffusioncoefficients determined in accordance with the method of the instantinvention are generally observed to be in good agreement withexperiment. Additionally, the instant system and method has demonstratedsubstantial sensitivity to the parameters of the domains (e.g., pores)within the PEM material where proton conduction generally occurs;including: sidechain length, length of pore, radius of pore, anddistribution of sulfonic acid fixed sites within the pore.

Moreover, the instant invention also discloses a method for computingthe permittivity of water as a function of pore parameters within PEMmaterials based on equilibrium statistical mechanics. The disclosedmodel has been applied to described inter alia the decrease in thedielectric constant of water molecules as they move from the center ofthe pore towards the wall of the pore. Results from the methodsdescribed herein (e.g., the modeling of proton transport and dielectricsaturation in accordance with various embodiments of the presentinvention) suggest that ionomeric PEM's exhibiting similar hydratedmorphologies (i.e., NAFION®, PEEKK and the like) will exhibit thehighest conductivities for fixed sites generally distributed in asubstantially homogeneous manner over the pore walls.

Optimized geometries for the chemical species of interest weredetermined by means of both ab initio Hartree-Fock theory and secondorder Møller-Plesset electron correlation corrections, and densityfunctional theory with Becke's three parameter hybrid method. Arepresentative custom-designed ionomer is disclosed as comprising achemical structure derived from a series of ab initio molecular basedcalculations and simulations; namely, a polymer electrolyte membranematerial having a novel sidechain for preventing or otherwiseameliorating increases in electro-osmotic drag when the materialexperiences increased swelling upon permeation with, for example,aqueous methanol. In one representative aspect, the present inventioninvolves the tailoring of the sidechain (through alteration of thechemical functionalization of the sidechain) such that hydrophilicity ofthe intermediate portion of the sidechain is increased beyond thattypically found in presently available PEM materials.

One representatively practical example of the present invention involvesthe incorporation of an ether oxygen (or oxygens) along the length ofthe sidechain of NAFION to introduce or otherwise improve hydrophilicity(e.g., the ability to form hydrogen bonds with water molecules). Anotherexample of how the structure of NAFION may be altered is disclosed ascomprising an alteration of the sidechain where the groups vicinal,geminal, etc. to the ether oxygen are changed from CF₂ to CH₂. Thedisclosed system and method may be readily and more generally adaptedfor use in the optimization and/or chemical functionalization of anyionomeric material, whether now known or otherwise hereafter describedin the art. The disclosed alteration of NAFION is shown inter alia asdramatically increasing the tendency of the ether oxygen to form ahydrogen bond with, for example, a water molecule.

The increased differential hydrophilicity of the sidechain generallymanifests itself as a representatively practical benefit, for example,when the ion-containing domains or pores swell by increasing theprotrusion of the sidechains within the pores. Characterization of thepermittivity of the water in the pores, in accordance with the methodsdisclosed in the instant invention, demonstrates that with increasedprotrusion of the sidechains, the transport of water by the protoniccurrent is inhibited, thereby preventing or otherwise amelioratingincreases in electro-osmotic drag. In addition, because thehydrophilicity of the sidechains may be custom designed so as to makethe terminal part more hydrophilic than along the length of thesidechain, the conductivity of the membrane at lower waterconcentrations generally will not be reduced with the increasedhydrophilicity of the sidechain.

Additional advantages of the present invention will be set forth in theDetailed Description which follows and may be obvious from the DetailedDescription or may be learned by practice of exemplary embodiments ofthe invention. Still other advantages of the invention may be realizedby means of any of the instrumentalities, methods or combinationsparticularly pointed out in the Claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Representative elements, operational features, applications and/oradvantages of the present invention reside inter alia in the details ofconstruction and operation as more fully hereafter depicted, describedand claimed—reference being had to the accompanying drawings forming apart hereof, wherein like numerals refer to like parts throughout. Otherelements, operational features, applications and/or advantages willbecome apparent to skilled artisans in light of certain exemplaryembodiments recited in the detailed description, wherein:

FIG. 1 generally depicts a schematic for a conventional Direct MethanolFuel Cell (DMFC) in accordance with the prior art;

FIG. 2 generally depicts an integrated fuel cell assembly in accordancewith the prior art;

FIG. 3 generally depicts an energy-minimized chemical structure for amonomeric moiety of NAFION 117 in accordance with the prior art;

FIG. 4 representatively illustrates an energy-minimized chemicalstructure for a monomeric moiety corresponding to a novel ionomericcompound in accordance with an exemplary embodiment of the presentinvention;

FIG. 5 representatively illustrates the oxygen-oxygen distance for anenergy-minimized interaction between a water molecule anddi-trifluoromethane ether;

FIG. 6 representatively illustrates the oxygen-oxygen distance for anenergy-minimized interaction between a water molecule and dimethylether;

FIG. 7 representatively illustrates the computed and experimental protondiffusion coefficients for NAFION 117 and 65% sulfonated PEEKK as afunction of hydration water content in accordance with one exemplaryaspect of the instant invention;

FIG. 8 representatively illustrates the relative permittivity of waterin NAFION 117 as a function of the radial distance from the pore walland hydration water content in accordance with another exemplary aspectof the instant invention; and

FIG. 9 representatively illustrates the parametric sensitivity of protondiffusion as a function of the length of sidechain pore intrusion inaccordance with yet another embodiment of the present invention.

Those skilled in the art will appreciate that elements in the Figuresare illustrated for simplicity and clarity and have not necessarily beendrawn to scale. For example, the dimensions of some of the elements inthe Figures may be exaggerated relative to other elements to helpimprove understanding of various embodiments of the present invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

The following descriptions are of exemplary embodiments of the inventionand the inventor's conception of the best mode and are not intended tolimit the scope, applicability or configuration of the invention in anyway. Rather, the following description is intended to provide convenientillustrations for implementing various embodiments of the invention. Aswill become apparent, changes may be made in the function and/orarrangement of any of the elements described in the disclosed exemplaryembodiments without departing from the spirit and scope of theinvention.

Various representative implementations of the present invention may beapplied to any composition of matter, system and/or method for thecharacterization of proton dissociation and transport as they may relateto hydrophilic components of hydrated ionomeric materials. A detaileddescription of an exemplary application, namely a composition of matterand a method for altering the differential hydrophilicity of thesidechain of NAFION is provided as a specific enabling disclosure thatmay be readily generalized by skilled artisans to any application of thedisclosed system and method in accordance with various embodiments ofthe present invention. Moreover, skilled artisans will appreciate thatthe principles of the present invention may be employed to ascertainand/or realize any number of other benefits associated with thecharacterization of proton dissociation and transport in PolymerElectrolyte Membrane materials such as, but not limited to: reduction ofelectro-osmotic drag; improvement of fuel cell efficiency; reduction ofdevice weight and/or form-factor; and any other applications or benefitswhether now known or hereafter developed or otherwise described in theart.

Fuel Cells

In the broadest sense, a fuel cell may be generally characterized as anydevice capable of converting the chemical energy of a supplied fueldirectly into electrical energy by electrochemical reactions. Thisenergy conversion corresponds to a free energy change resulting from theoxidation of a supplied fuel. As schematically depicted, for example, inFIG. 1, a typical prior art fuel cell consists of an anode (e.g., ‘fuelelectrode’) 130 that provides a reaction site to generate electrons anda cathode (e.g., ‘oxidant electrode’) 135 to reduce spent fuel ions inorder to produce a voltage drop across, for example, an externalcircuit. The electrodes 130, 135 are generally ionically porouselectronic conductors that include catalytic properties to providesignificant redox reaction rates. At the anode 130, incidenthydrogen-containing fuel 162 (e.g., methanol 185) catalytically ionizesto produce protons 175 (e.g., electron-deficient hydrogen nuclei) andelectrons. At the cathode 135, incident oxygen gas 165 catalyticallyreacts with protons 175 migrating through the electrolyte 140 andelectrons from the external circuit to produce water 180 as a byproduct.Depending on various operational parameters of the fuel cell, byproductwater 180 may remain in the electrolyte 140, thereby increasing thevolume and diluting the electrolyte 140, or may be discharged from thecathode 135 as vapor 178.

The anode 130 and cathode 135 are generally separated by anion-conducting electrolytic medium 140 (i.e., PEM's or alkali metalhydroxides such as, for example: KOH, NaOH and/or the like). In earlyfuel cell experiments, hydrogen and oxygen were introduced intoseparated compartments while the electrodes where conductively coupledby an external circuit to power a load where electrical work could beaccomplished. In the external circuit, electric current is generallytransported by the flow of electrons, whereas in the electrolyte 140,current is generally transported by the flow of ions. In theory, anychemical substance capable of oxidation (i.e., hydrogen, methanol,ammonia, hydrazine, simple hydrocarbons, etc.) which may be suppliedsubstantially continuously may be used as galvanically oxidizable fuelat the anode. Similarly, the oxidant (i.e., oxygen, ambient air, etc.)may be selected to be any substance that can oxidize spent fuel ions ata sufficient rate to maintain a suitable voltage drop across thecircuit.

The free energy of reaction ΔG of a fuel cell is given as ΔG=ΔE+ΔH,where ΔE is the energy available to accomplish electrical work and ΔH isthe energy liberated from the reaction to raise the temperature of thefuel cell and the surroundings. In typical fuel cell applications, theheat liberated to the fuel cell's surroundings is much less than theenergy available to accomplish electrical work; which may be expressedas: ΔH

ΔE.

For example, where $Q_{FuelCell} = \frac{E_{Electrical}}{E_{Chemical}}$represents the efficiency of converting chemical potential energyE_(Chemical) directly to electrical energy E_(Electrical), typicalhydrogen/oxygen fuel cell efficiencies on the order of Q_(FuelCell)=0.65to about Q_(FuelCell)=0.80 have been observed. These values are nearlytwice those of indirect heat-exchange power conversion methods, whichmay be expressed by the following relation:Q_(FuelCell)=2Q_(HeatExchange)

where the heat-exchange efficiency is given as$Q_{HeatExchange} = {\frac{E_{Combustion}}{E_{Chemical}} \times \frac{E_{Electrical}}{E_{Combustion}}}$

The factor E_(Combustion)/E_(Chemical) represents the componentefficiency of converting chemical potential energy into heat (i.e., thecombustion of fossil fuels) and the factor E_(Electrical)/E_(Combustion)represents the component efficiency of converting heat into electricalenergy; for example, steam-driven turbo-electric power.

Accordingly, fuel cell operation is intrinsically more efficient thanmethods employing heat-exchange power conversion. Moreover, otherrepresentative benefits of fuel cells include higher energy densities,quiet operation and the lack of recharging and/or electrode replacementrequirements.

Standard batteries have generally dominated the currently availablechoices for portable power storage solutions for consumer-levelelectronic equipment. Some of the disadvantages associated with standardbatteries, however, are that they generally provide power for arelatively short duration of time and thereafter require recharging orreplacement. Fuel cells, on the other hand, have many of theconsumer-oriented features typically associated with standard batteries(i.e., providing quiet power in a convenient and portable package) inaddition to other representative advantages including, for example, longusage lifetimes and the ability to be fueled with liquid or gaseouscompounds rather than ‘solid fuels’ in contrast to conventionalbatteries.

One class of fuel cells currently under development for consumer use isthe hydrogen fuel cell, wherein hydrogen-rich fuels (i.e., hydrogen,methanol 162, methane, etc.) are used to fuel the redox reaction. Asfuel is oxidized at the anode 130, protons 175 pass through the cell forreduction at the cathode 135. In the case of using aqueous methanol 162as the fuel for example, carbon dioxide 168 is formed as a byproduct atthe anode 130. Free electrons from the external circuit then affectreduction of oxygen 165 at the cathode 135. The reduced oxygen 165 thencombines with protons to produce water 178. Conventional fuel celldevices may also include charge collectors 190, 195 for effectivedelivery of electric current to, for example, an external circuit.

One process for fueling a hydrogen cell comprises that of ‘directoxidation’ methods. Direct oxidation fuel cells generally include fuelcells in which an organic fuel is fed to the anode for oxidation withoutsignificant pre-conditioning or modification of the fuel. This isgenerally not the case with ‘indirect oxidation’ (e.g., “reformer”) fuelcells, wherein the organic fuel is generally catalytically reformed orprocessed into organic-free hydrogen for subsequent oxidation. Sincedirect oxidation fuel cells do not generally require fuel processing,direct oxidation provides substantial size and weight advantages overindirect oxidation methods. Exemplary prior art for direct and indirectfuel cells has been previously disclosed and may be compared, forexample, in U.S. Pat. Nos. 3,013,908; 3,113,049; 4,262,063; 4,407,905;4,390,603; 4,612,261; 4,478,917; 4,537,840; 4,562,123; 4,629,664 and5,599,638.

Another well-known type of fuel cell component is known as a‘membrane-electrode assembly’ (MEA), as described for example in U.S.Pat. No. 5,272,017 issuing on Dec. 21, 1993 to Swathirajan. Oneexemplary embodiment of such an MEA component includes a DMFC asgenerally depicted, for example, in FIG. 2. The illustrated DMFC MEAcomprises a thin, proton-transmissive, solid polymer-membraneelectrolyte 200 having an anode 210 a on one of its faces and a cathode210 b on an opposing face. The DMFC MEA anode 210 a, electrolyte 200 andcathode 201 b may also be sandwiched between a pair of chemicallypermeable elements 220 a, 220 b. This assembly may be further sandwichedbetween a pair of collector elements 230 a, 230 b which serve as currentcollectors for the anode 210 a and cathode 210 b respectively andcontain appropriate channels and/or openings 235 a, 235 a for thedistribution of fuel (i.e., aqueous methanol—in the case of a DMFCdevice) and oxidant reactants (i.e., oxygen) over the surfaces of thecorresponding electrode catalysts. In practice, a number of these unitfuel cells may be stacked or grouped together to form a ‘fuel cellstack’. The individual cells may be electrically connected in series byabutting the anode current collector of one cell with the cathodecurrent collector of a neighboring unit cell in the stack.

As the DMFC anode is fueled with a mixture of methanol and water, theoxidation reaction generally proceeds in what is believed to comprisethree steps: (1) methanol oxidizes to methanal (e.g., formaldehyde),releasing two electrons; (2) methanal oxidizes to methanoic acid (e.g.,formic acid), releasing two electrons; and (3) methanoic acid oxidizesto carbon dioxide, releasing another two electrons. In variousembodiments of exemplary DMFC's, the oxidation reaction may be startedat any point in the multi-step series since the two intermediates(methanal and methanoic acid) are generally readily obtainable. Thefirst oxidative step (methanol to methanal) is generally regarded as therate-determining step of the overall reaction given spectroscopicstudies indicating that methanal and methanoic acid appear in relativelylow concentrations. This would suggest that these intermediates arerapidly oxidized and accordingly, the reaction steps corresponding totheir oxidative consumption would be expected to have larger kineticrate constants. The net anode reaction for a direct methanol-fueleddevice is therefore typically given as:CH₃OH+H₂O→6H⁺+6e⁻+CO₂

Generally, the current produced by a DMFC is proportional to the netreaction rate, wherein one ampere corresponds to approximately 1.04 E 18reactions per second. As aqueous methanol is oxidized at the anode,electrons are liberated to flow through an external circuit to power aload where electrical work may be accomplished. Protons migrate throughthe proton-transmissive electrolytic membrane where they subsequentlyare combined with oxygen that has been reduced with incoming electronsfrom the external circuit with water formed as a result.

Diffusive Transport—A Qualitative Description

Given a thin barrier of infinite permeability and cross-sectional area Athat extends from x to x+l (where l represents the thickness of thebarrier), the volume of the barrier may be expressed as V=Al. Let theconcentration at point x of particles G be [G] at time t. Accordingly,the number of particles that enter the barrier per unit time is JA whereJ is the particle flux. Therefore, the rate of increase in molarconcentration inside the barrier due to the incoming particle flux is${\frac{\partial\lbrack G\rbrack}{\partial t}}_{x} = {\frac{JA}{Al} = {\frac{J}{l}.}}$Consider also an out-bound flux of particles at the x+l surface of thebarrier which may be similarly derived as${\frac{\partial\lbrack G\rbrack}{\partial t}}_{x + l} = {\frac{J^{\prime}A}{Al} = {\frac{J^{\prime}}{l}.}}$Therefore, the net time rated change of concentration (e.g., the‘concentration velocity’) may be expressed as:$\frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}t} = \frac{J - J^{\prime}}{l}$

Suppose: (1) that the flux of particles J diffusing inside the barriercomprises motion in response to a thermodynamic force F arising from aconcentration gradient; (2) that the particles reach a steady-statedrift speed s when the thermodynamic force F is matched by the viscousdrag; (3) that the drift speed s is proportional to the thermodynamicforce F; (4) that the particle flux J is proportional to the driftspeed; and (5) that the thermodynamic force F is proportional to thespatial concentration gradient d[G]/dx. The resulting chain ofproportionalities J ∝ s, S ∝ F, and$F \propto \frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}x}$implies that the particle flux J is proportional to the concentrationgradient d[G]/dx, which will be apparent to skilled artisans ascorresponding to ‘Fick's First Law of Diffusion’. The constant ofproportionality is given as the diffusion coefficient D in the equation$J = {D\frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}x}}$for diffusion restricted to a single dimension x. Therefore, theexpression J-J′ taken from the expression for the diffusiveconcentration velocity becomes${D\frac{\mathbb{d}\lbrack G\rbrack^{\prime}}{\mathbb{d}x}} - {D{\frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}x}.}}$Substitution of the linear accumulation of particle concentration overthe thickness of the membrane yields${J - J^{\prime}} = {{D\frac{\mathbb{d}}{\mathbb{d}x}\left( {\lbrack G\rbrack + {\frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}x}l}} \right)} - {D\frac{\mathbb{d}\lbrack G\rbrack}{\mathbb{d}x}}}$which further reduces to${J - J^{\prime}} = {{Dl}{\frac{\mathbb{d}^{2}\lbrack G\rbrack}{\mathbb{d}x^{2}}.}}$This expression may then be substituted back into the concentrationvelocity expression to yield:${\frac{d\lbrack G\rbrack}{dt} = {\frac{J - J^{\prime}}{l} = {{D\frac{d^{2}\lbrack G\rbrack}{{dx}^{2}}} = {D{\nabla_{x}^{2}\lbrack G\rbrack}}}}},$

which will be apparent to skilled artisans as the time dependentdiffusion equation according to ‘Fick's Second Law of Diffusion’ andrelates the concentration velocity at any point to the spatial variationof the concentration at that point. More generally, this may beappreciated as a physical basis for the typically observed behavior ofdiffusing chemical species translating away from areas of relative highconcentration to areas of relative lower concentration (e.g., “movingdown the concentration gradient”).

Next, consider the time dependence of the partial molecular pressure pof effusing particles G from a container of given volume V. The ‘IdealGas Law’ PV=nRT, which for molecular-scale systems rather than for largeaggregates of particles (i.e., moles of molecules), becomes pV=nkTwherein:

p is the partial molecular pressure;

V is the volume of the container providing spatial boundary conditions;

n is the number of particles;

k is the Boltzmann constant; and

T is the temperature.

Solving for the partial pressure yields $p = {\frac{nkT}{V}.}$After taking the partial derivative with respect to time at constanttemperature and volume, the following expression for the pressurevelocity may be obtained:$\left. \frac{\partial p}{\partial t} \right)_{T,V} = {\frac{\partial\left( \frac{nkT}{V} \right)}{\partial t} = {\frac{kT}{V}{\frac{\partial n}{\partial t}.}}}$

For effusing particles that are not replenished over time as theparticles escape, the time-rated change of the number of particles isgiven as ${\frac{\partial n}{\partial t} = {{- Z_{w}}A_{o}}},$where Z_(w) is the collisional frequency associated with the mean freepath of the particles and A_(o) is the area of the opening that theeffused particles have available to escape from. The collisionalfrequency is related to the partial pressure of the particles p, themass of the particles m and the temperature of the system T by theequation $Z_{w} = {\frac{p}{\sqrt{2\quad\pi\quad{mkT}}}.}$Substitution of this relation back into the expression for the pressurevelocity yields$\frac{\partial p}{\partial t} = {\frac{- {pA}_{0}}{V}\sqrt{\frac{kT}{2\quad\pi\quad m}}}$which integrates over time to${p = {p_{0}{\mathbb{e}}^{\frac{- t}{\tau}}}},$where $\tau = {\frac{V}{A_{0}}{\sqrt{\frac{2\quad\pi\quad m}{kt}}.}}$From this expression for the effusive pressure velocity, the followingmay generally be observed: (1) if the particle matter is notreplenished, the pressure decreases exponentially to zero; (2) thepressure velocity is faster with increasing temperature and slower withdecreasing temperature; (3) the pressure velocity is slower with heavierparticles and faster with less massive particles; (4) the pressurevelocity is faster with increasing surface area of the effusiveopening(s) and slower with decreased surface area; and (5) the pressurevelocity is slower with increasing volume of the effusive container andfaster with increasing volume.

At constant temperature, the time derivative of the expression for thepartial pressure $p = \frac{nkT}{V}$becomes:$\left. \frac{\partial p}{\partial t} \right)_{T} = {{{kT}\frac{\partial\left( \frac{n}{V} \right)}{\partial t}} = {{kT}{\frac{\partial\lbrack G\rbrack}{\partial t}.}}}$

Therefore, substituting the expression corresponding to Fick's SecondLaw of Diffusion for the concentration velocity previously derived, thegeneralized expression for the pressure velocity of particles diffusingin three dimensions in a barrier of infinite permeability as a functionof concentration of the particles [G] may be represented as:$\left. \frac{\mathbb{d}p}{\mathbb{d}t} \right)_{T} = {{{- {kTD}_{G}}{\nabla^{2}\lbrack G\rbrack}} = {- {{{{kTD}_{G}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)}\lbrack G\rbrack}.}}}$

If, however, the barrier is assumed to have finite diffusivepermeability, an additional diffusion coefficient {circumflex over(D)}_(Ψ(a,b,c . . . )) may be included to account for variousbarrier-dependent permeability metrics such as, for example: non-uniformporosity; anisotropic transport along different dimensions;hydrophobicity; hydrophilicity; barrier/membrane/capillary defects; etc.

As enabling disclosure for a representative embodiment directed to anexemplary DMFC system in accordance with one aspect of the presentinvention is developed, it may be convenient to consider the followingqualitative expression for protons diffusing through a membrane (orotherwise porous barrier) Ψ:$\left. \frac{\mathbb{d}p}{\mathbb{d}t} \right)_{\Psi,H^{+}}^{diffusion} = {{- {{kT}\left( {{\hat{D}}_{\Psi({a,b,{c\ldots}}\quad)}D_{H^{+}}} \right)}}{\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)\left\lbrack H^{+} \right\rbrack}}$

Upon inspection, this expression relates the protonic concentration atany point within a membrane (or otherwise porous barrier) Ψ to thethree-dimensional variation of proton concentration at that point; whichis to say that protons will diffuse through a porous barrier so as tomove down the protonic concentration gradient from volume elementscomprising higher proton concentrations to volume elements comprisingrelative lower proton concentrations.

Substitution of A_(Ψ)l for the volume V in the effusion equation$\frac{\mathbb{d}p}{\mathbb{d}t} = {{\frac{- {pA}_{0}}{V}\sqrt{\frac{kT}{2\pi\quad m}}\quad{yields}\quad\frac{\mathbb{d}p}{\mathbb{d}t}} = {\frac{- {pA}_{0}}{A_{\Psi}l}{\sqrt{\frac{kT}{2\pi\quad m}}.}}}$If the ratio of the area of the membrane openings to the aggregatesurface area of the membrane normal to the effusing particle transportpath is taken to be a dimensionless quantity θ corresponding to theporosity of the membrane at the surface of effusion, a compositeexpression for protons effusing from a membrane (or otherwise porousbarrier) surface becomes:$\left. \frac{\mathbb{d}p}{\mathbb{d}t} \right)_{\Psi,H^{+}}^{effusion} = {{- p_{H^{+}}}\theta_{\Psi}\sqrt{\frac{kT}{2\pi\quad m_{H^{+}}l_{\Psi}^{2}}}}$

and relates the vapor pressure of protons that diffuse through thebarrier to reach the exterior surface of the porous barrier Ψ to: theback-side pressure of protons p_(H) ₊ ; the temperature T; the mass ofthe protons m_(H) ₊ ; the thickness of the barrier l; and the porosityof the barrier θ_(Ψ). Accordingly, protons will generally be observedto: (1) effuse more rapidly at higher operating temperatures and moreslowly at lower temperatures; (2) effuse more rapidly with increasedback-side pressure of protons and more slowly with decreased back-sidepressure; (3) effuse more rapidly with membranes having increasedporosity and more slowly with decreased porosity; and (4) effuse morerapidly with porous barriers having decreased linear transport distances(e.g., thinner membranes) and more slowly with increased transportdistances.

Polymer Electrolyte Membranes

Although ion-containing polymers have been around for some time, thepast fifteen years have witnessed much interest in the literature. Mostof the research effort on these ionomers have generally been devoted toa relatively small number of materials—notably: ethylenes, styrenes,rubbers, and those based on, for example, poly-tetrafluoroethylene. See,for example, Butler, G. B. et al., JMS-Rev. Macromol. Chem. Phys.C34(3), 325-373 (1994). NAFION® (available from DuPont MicrocircuitMaterials, E. I. Du Pont de Nemours and Company, 14 T.W. AlexanderDrive, Research Triangle Park, N.C., USA) is one such example of apoly-tetrafluoroethylene-based ionomer. From its development in the1960's, NAFION has found applications in liquid and gas separations,fuel cells as well as in the chlor-alkali industry. The diversity ofapplications for NAFION may be due in no small part to its thermalstability, chemical resistance, ion-exchange properties, selectivity,mechanical strength and insolubility in water. NAFION's ability to formor otherwise provide wide, well-separated, highly-branched channels withgood channel connectivity and relatively small SO₃ ⁻/SO₃ ⁻ separation,in addition to a pK_(a)˜−6, generally recommends itself to integrationwith a variety of fuel cell applications.

NAFION is a perfluorinated polymer that contains small proportions ofsulfonic or carboxylic ionic functional groups. The general polymericstructure of NAFION may be conveniently represented as:

In terms of macroscopic structure, NAFION is relatively complex.Although the exact structure has not been characterized, several modelshave been proposed since the early 1970's to describe the way in whichionic groups aggregate within the NAFION polymer. These models generallyinclude: The Mauritz-Hopfinger Model, The Yeager Three-Phase Model, TheEisenberg Model of Hydrocarbon lonomers, and The Gierke Cluster NetworkModel. See, for example, Robertson, M. A. F., Ph.D. Thesis, Universityof Calgary (1994). A common objective of these models is to predict thefundamental feature of unique equilibrium ionic selectivities as well asthe ionic transport properties of perfluorinated ionomer membranes.Ibid.

As a result of electrostatic interactions, these ionic groups tend toaggregate to form tightly packed regions referred to as clusters. Thepresence of these electrostatic interactions between the ions and theion pairs generally enhances the intermolecular forces, thereby exertinga significant effect on the properties of the parent polymer.

The Yeager Three Phase Model is a phenomenological model based on athree-phase clustered system with interconnecting channels within thepolymer. The three regions consist of a fluorocarbon backbone (some ofwhich is microcrystalline), an interfacial region of relatively largefractional void volume (containing some pendant sidechains, some water,and sulfate or carboxylic groups and counter ions which are generallynot in clusters), and the clustered regions where the majority of theionic exchange sites, counter ions, and sorbed water exists. See, forexample, Brookman, P. J.; Nicholson, J. W. in: Developments in IonicPolymers, vol. 2, 269-283, Elsevier Applied Science Publishers: London(1986).

From experimental means, such as for example, small-angle x-rayscattering (SAXS), it has been observed that the phase-separatedmorphology is on the order of about 40-50 Å Bragg spacing; however, uponhydration, NAFION can increase its dry weight by as much as 50 percentor more depending upon equivalent weight, counter ion and temperature.Upon hydration, however, cluster diameter and the number of exchangesites are thought to increase, leading to fewer, larger clusters. Ibid.

NAFION, with its existing ionic clusters and postulated inter-clusterchannels, serves not only as a stable platform or template, but also asa catalyst. The sulfonate exchange sites in the ionomer are quiteacidic. Therefore, the clusters in NAFION generally serve as reactionvessels in which future polymerizations may occur without the additionof an external catalyst. The pre-existing morphology of NAFION, asdiscussed vide supra, has a direct influence on the in situ morphologyof any inorganic phase in view of the fact that the clusters are only onthe order of about 40-50 Å in size. Therefore, one can generate distinctordered structures in the clusters and form a network between clustersusing the short channels that connect the aggregates. Upon doing so, theoriginal ionomer properties may be altered and tailored to specific usesand needs, such as for example, specific gas and liquid separationsand/or fuel cell applications.

Other ionomeric materials that may find improved derivatization inaccordance with various exemplary embodiments of the instant inventioninclude:

Dow Membrane;

Sulfonyl Imide Monomer; and

PEEKK.

PEEKK is a sulfonated polyether ketone capable of forming or otherwiseproviding narrower, less-separated channels with somewhat larger SO₃⁻/SO₃ ⁻ separation as compared with NAFION. Additionally, PEEKK has amuch lower pK_(a) on the order of about −1.

Distinctly different efforts into the design of advanced andcost-effective membranes include: (1) aromatic backbone polymers, i.e.polyether ketones (PEEKK, PEEK, etc.); (2) the inclusion of smallinorganic particles like silica, or zirconium phosphates andsulfophenylphosphates within the membrane; (3) acid/base blending orcovalent cross-linking of polymers; and (4) the complexation of basicpolymers (i.e., polybenzimidazole) with oxo-acids (i.e., phosphoricacid). The PEEKK membranes offer definite cost and stability advantagesover Nafion membranes, but exhibit substantially lower conductivity atthe lower water concentrations. The membranes in (2) and (3) exhibitincreased thermal stability (up to 140° C.) and reduced swelling andmethanol and water crossover, but at a penalty in terms of conductivityand mechanical stability. Finally, the membranes with immobilized acidgenerally demonstrate conductivities as high as those seen in thehydrated systems but with substantially reduced methanol crossover.

Although there has been substantial work in the synthesis and testing ofvarious membranes, many performance features are not well understood.Equally deficient is a fundamental, molecular-based understanding of themechanisms of proton and water transport as a function of membranemorphology and hydration. Clearly, success in the design of novelmembranes possessing many of the characteristics required for use in acommercial fuel cell will require the fundamental physical andmechanistic insight generated from molecular modeling studies.

The hydrated morphology of the sulfonic acid based ionomers has a directbearing on the transport of protons and water in the membrane. Theperfluoro polymers combine relatively high hydrophobicity of thepolytetrafluoroethylene (PTFE) backbone with the relatively highhydrophilicity of the sulfonic acid functional groups in a singlemacromolecule. In the presence of water a two-phase system formsconsisting of a network of water containing clusters or pores surroundedby, for example, the PTFE medium. The latter typically provides thestructural and thermal stability of the membrane and is also responsiblefor the immobilization of the dissociated sulfonic acid groups (—SO₃ ⁻);hence, referred to as “fixed sites”. It is within the hydrophilicdomains that the transport of water and protons typically occurs. Aninterfacial region, therefore, exists consisting of solvatedsulfonate-terminated perfluoroether sidechains. This interface separatesa water region in the central portion of the pore that is ‘bulk-like’,from the PTFE backbones. The nature and character of the water in thepores, however, was previously not well understood in the prior art.

In the modeling work of Eikerling et al. the mobility of the protons wasthought to occur via two mechanisms: a surface mechanism where protontransport proceeds along the array of acid groups (i.e., via structurediffusion) along the interface and a bulk mechanism where the protonsare transported with the Grotthuss mechanism. See, for example, M.Eikerling at al., J. Phys. Chem. B, 105, 3646 (2001). In addition,Eikerling attempted to model membrane morphological aspects includingconnectivity of the hydrophilic domains, orientation of the pores in thenetwork and evolution of the pore volume with water uptake.

The hydrated morphologies and consequent function of polyether ketonemembranes is somewhat different from the perfluoro polymers. Small angleX-ray scattering (SAXS) experiments suggest that there is a lesspronounced separation of the hydrophobic and hydrophilic domains thanobserved in NAFION membranes. This along with the greater rigidity ofthe aromatic backbone of the polymer result in narrower water filledpores. In addition, results from pulsed-field gradient NMR measurementsindicated that the electro-osmotic drag and water permeation is lower inthe PEEKK membranes.

Although previous experimental investigations provide a qualitativeunderstanding of the function of the sulfonic acid based ionomers(perfluoro and aromatic), the specific details of how the molecularstructure and hydrated morphologies connect with the transport ofprotons and water through the membranes was not characterized. Thedevelopment of novel materials for application in fuel cells willgenerally require a fundamental understanding of the function ofexisting materials. Thus, in one exemplary and representative aspect,the present invention provides inter alia a modeling system and methodfor characterizing the hydration and acid dissociation in these classesof ionomers with the aim of connecting fundamental studies with thefunction of hydrated membranes.

With any ionomer, it may well be currently impossible to treat theentire polymer in an ab initio manner (i.e., a full electron treatmentwith molecular orbital theory). Treatment of the polymer with empiricalor even semi-empirical methods, while computationally feasible, willgenerally yield conformational results for the polymer interacting with,water that are largely incorrect due to the approximations employed forthe force fields. Accordingly, for any ionomer modeled in accordancewith the method described herein, the smallest monomeric sub-unit of thepolymer that contains the essential, membrane specific molecularcomponents may be considered. For NAFION, this sub-unit may be taken tobe trifluoromethane sulfonic acid (triflic acid) and for the PEEKKmembranes, para-toluene sulfonic acid for the characterization of, forexample, proton dissociation. In terms of characterization of propertiesof the sidechain, the monomeric unit generally depicted in FIG. 3 may beused, for example, to model NAFION. Those skilled in the art willappreciate that various substantially canonical representations ofchemical structures and/or sub-structures may be effectively employed toarrive at substantially similar results in accordance with the instantinvention and that such representations are considered within the scopeand ambit of the present invention.

In the first stage of modeling, the perfluoro and aromatic ionomerelectronic structure calculations of the corresponding acid withexplicit water molecule solvation are treated. These computations aregenerally used to determine minimum energy conformations revealing interalia fundamental characteristics of the acid dissociation and localproton transport dynamics. In addition they also reveal informationconcerning the shielding or screening afforded by water molecules of,for example, the first hydration shell.

Molecular information may thereafter be obtained that, when combinedwith experimental studies of the hydrated morphology of the polymer,generally provide a set of parameters for implementation in a water andproton transport model. This is given as the second stage of themodeling protocol and is generally based on the computation of, forexample, the proton friction and diffusion coefficients within a PEMpore using a non-equilibrium statistical mechanical framework. Analgorithm of this model is presented vide supra. Together, the molecularstructure and transport modeling methods provide the means forconnecting the molecular scale information of the polymer with themacroscopic (and therefore experimentally measurable) transportproperties of the membrane. Of substantial significance to how the priorart has been improved by the instant invention is the fact that bridgingof the different length and time scales may be accomplished withoutresorting to any ‘fitting’ or adjustable parameters—indicating interalia the improved predictive capability of the system and methoddescribed herein.

Modeling and Computational Methods

Electronic Structure Calculations

All ab initio self-consistent-field (SCF) molecular orbital calculationswere performed using the G98 suite of programs (Gaussian98, revisionA.9; available from Gaussian Inc., Pittsburgh, Pa., USA). Full geometryoptimizations, using conjugated gradient methods to accelerateconvergence (see, for example, H. B. Schlegel, J. Comp. Chem. 3, 214(1982)) were undertaken on the acids (i.e., CF₃SO₃H and CH₃C₆H₄SO₃H)without symmetry constraints using Hartree-Fock theory with the6-31G(d,p) split valence basis set. See, for example, P. C. Hariharan,J. A. Pople, Theo. Chim. Acta. 28, 213 (1973). The HF/6-31G(d,p) minimumenergy conformations were then refined with density functional theorywith Becke's 3 parameter functional (B3LYP) with the same basis set.See, for example, A. D. Becke, J. Chem. Phys. 98, 5648 (1993). Watermolecules were then systematically added to the B3LYP/6-31G(d,p) minimumenergy structure to obtain successively larger solvation clusters of theacid (i.e., SO₃H+nH2O; 1≦n≦6); and the same discrete optimizationprotocol conducted. Electrostatic potential derived atom centeredpartial charges were obtained for the B3LYP/6-31G(d,p) minimum energyclusters according to the ChelpG scheme. See, for example, C. M.Breneman, K. B. Wiberg, J. Comp. Chem. 11, 361 (1990).

Proton Transport Model

Mathematical details of our model in the form of a derivation from firstprinciples were present earlier. See, for example, S. J. Paddison atal., “Proton Conducting Membrane Fuel Cells II”, Electrochemical Soc.Proc. Series, Pennington, N.J. 98-27, 106 (1999); S. J. Paddison et al.,J. Electrochem. Soc. 147, 617 (2000); and S. J. Paddison at al., J.Chem. Phys. 115, 7753 (2001). In general, the molecular structure of thehydrated polymer is connected with the macroscopic (e.g., measurable)quantity of the proton diffusion coefficient. Factors affecting thecoupled transport of a proton and a water molecule (i.e., a hydroniumion, designated subscript a in the expressions vide infra) are examinedin a hydrated pore/channel of a PEM ex situ of a fuel cellconfiguration.

The Einstein relation $D_{\alpha} = \frac{kT}{\zeta_{\alpha}}$generally establishes the inverse relationship of the diffusioncoefficient with the friction coefficient. While the Stokes relationζ=6πηα is commonly used to compute friction coefficients for macroscopicobjects moving in, for example, viscous media (which in combination withthe Einstein relation produces the Stokes-Einstein formula). In oneembodiment of the present invention, a method of non-equilibriumstatistical mechanics is employed to compute the average forceexperienced by, for example, a hydronium ion moving in the pore of anionomeric PEM material, making use of the fundamental definition of thefriction coefficient <F_(α)>=−ζ·υ_(α), where υ_(α) is the velocity(assumed to be constant) of the hydronium ion. Accordingly, viacomputation of the average force, the diffusion coefficient may beevaluated.

The pore of the ionomeric PEM material is assumed to possess acylindrical geometry with length L and cross sectional radius R, filledwith N water molecules each possessing a dipole moment μ. Thedissociated sulfonic acid functional groups (—SO₃ ⁻) in the pore aremodeled as n radially symmetric, axially periodic arrays of fixed ions(i.e., point charges) each possessing a charge of e⁻. In accordance withone exemplary embodiment of the present invention, the average forceexperienced by a hydronium ion in such a pore may be calculated from thestatistical mechanical relation:<F _(α)>(r _(α))=ƒF _(α)(r _(α) , r)ρ(r _(α) , ρ, r)dr·dpwhere r_(α) denotes the position of the hydronium ion and the average(e.g., integration) is over the position r and conjugate momentum p ofall N water molecules of the net force on the hydronium weighted with aphase space distribution function corresponding to ρ(r_(α), p,r). Thisdistribution function may be generally obtained from the more generictime-dependent distribution function; a solution of the time evolutionor Liouville equation:${i\frac{\partial{\rho\left( {r_{\alpha},p,r,t} \right)}}{\partial t}} = {L_{o}{\rho\left( {r_{\alpha},p,r,t} \right)}}$where L_(o) is the Liouville operator for the pore system in accordancewith a representative aspect of the present invention with a coordinatereference system moving with constant velocity ν_(α). The Liouvilleoperator may be defined by the Poisson bracket L_(o)=i{Ho_(o)(r_(α), p ,r)}, where H_(o)(r_(α), p, r) corresponds to the Hamiltonian for thepore. The total energy of the pore will generally consist of the kineticenergy of all the water molecules and the net potential energy V(r_(α),r) due to two-body interactions of the water molecules, hydronium ion,and fixed sites according to, for example:${H_{o}\left( {r_{\alpha},p,r} \right)} = {{\sum\limits_{i = 1}^{N}\frac{{m\left( {v_{i} + v_{\alpha}} \right)}^{2}}{2}} + {V\left( {r_{\alpha},r} \right)}}$where m is the mass and ν_(i) is the velocity of the i^(th) watermolecule. The latter term here comprising, for example, the followingfour contributions: $\begin{matrix}{{V\left( {r_{\alpha},r} \right)} = {{- {\sum\limits_{i = 1}^{N}{\frac{\mu^{2}e^{2}}{48\pi^{2}ɛ^{2}{kT}}\frac{1}{{{r_{\alpha} - r_{i}}}^{4}}}}} + {\Psi_{o}{\cos\left( \frac{2\pi\quad{nz}_{\alpha}}{L} \right)}} +}} \\{{\sum\limits_{i < j}^{N}{\frac{2\mu^{4}}{3\left( {{4\pi} \in} \right)^{2}{kT}}\frac{1}{{{r_{i} - r_{j}}}^{6}}}} - {\sum\limits_{i = 1}^{N}{\frac{2{\pi\mu\Psi}_{o}n}{eL}{\sin\left( \frac{2\pi\quad{nz}_{i}}{L} \right)}}}}\end{matrix}$where ε is the permittivity of the water in the pore, k the Boltzmannconstant, T the temperature and Ψ_(o) the amplitude of the potentialenergy due to interaction of the hydronium ion with the —SO₃ groups.These respective contributions to the potential energy of the system aredue inter alia to: (1) interactions of the hydronium ion with the watermolecules; (2) interaction of the hydronium ion with the arrays of thefixed sites; (3) water-water interactions; and (4) interactions of thewater molecules with the fixed sites. Accordingly, a formal solution ofthe Louiville equation may be expressed as:ρ(r_(α), p, r, t) = 𝕖^(−𝕚  L_(o)t)ρ(r_(α), p, r, 0) = 𝕖^(−𝕚  L_(o)t)ρ_(eq)(r_(α), p, r)where ρ_(eq)(r_(α), p, r) may be understood as the distribution functionunder equilibrium conditions. A non-equilibrium stationary state (i.e.,in the moving ion frame of reference) and described by the distributionfunction in accordance with the average force equation may be obtainedin the limit of t→∞ in the expression for the formal solution of theLouiville equation vide supra. The total force required may bedetermined, for example, by taking the action of the Louiville operatoron the momentum of the hydronium ion. Combining these results, one mayobtain an expression for the scalar friction coefficient of thehydronium comprising at least four force-force correlation functions:$\begin{matrix}{\zeta_{\alpha} = {\frac{\beta}{3}{\int_{0}^{\infty}\left( {\left\langle {F_{\alpha\quad s}{\mathbb{e}}^{{- {\mathbb{i}}}\quad L_{o}t}F_{\alpha\quad s}} \right\rangle_{0} + \left\langle {F_{\alpha\quad s}{\mathbb{e}}^{{- {\mathbb{i}}}\quad L_{o}t}F_{p\quad s}} \right\rangle_{0} + \left\langle {F_{\alpha\quad p}{\mathbb{e}}^{{- {\mathbb{i}}}\quad L_{o}t}F_{p\quad s}} \right\rangle_{0} +} \right.}}} \\{\left. \left\langle {F_{\alpha\quad p}{\mathbb{e}}^{{- {\mathbb{i}}}\quad L_{o}t}F_{\alpha\quad s}} \right\rangle_{0} \right){\mathbb{d}t}}\end{matrix}$where $\beta = \frac{1}{kT}$and the forces F_(αs), F_(ps) and F_(αp) are generally experiencedbetween: (1) the hydronium ion and the water molecules; (2) the fixedsites and the water molecules; and (3) the hydronium ion and the fixedsites, respectively. In an exemplary and representative embodiment ofthe present invention, the later three terms are explicitly evaluated,taking their sum to be a correction ζ^((c)) to the friction coefficientof the proton in bulk water; for example:ζ^((c))=ζ₂+ζ₃+ζ₄

Examination of the first force-force correlation function ζ₁ indicatesinvolvement of the force the water exerts on the hydronium ion F_(αs),and so may be taken to correspond to either the friction coefficient ofa hydronium ion in bulk water calculated with the Stokes relation or thefriction coefficient of a proton in bulk water derived from experimentaldiffusion measurements. The choice of the numerical value of ζ₂ maydepend on the characteristics of water (i.e., in the pore) through whichthe proton moves and is discussed further vide infra.

Representative Results

As indicated vide supra (for example, from B3LYP/6-31G** minimum energycalculations for the monomeric acids) a single water molecule wasbrought in proximity to the sulfonic acid portion of the ionomericmolecule of interest with optimizations performed; first at theHF/6-31G** level and then at the B3LYP/6-31G** level of theory.Successive water molecules were then individually added without any biasor constraints as to the starting geometry prior to the optimization.

Minimum energy conformations (B3LYP/6-31G**) for triflic acid with thesuccessive addition of six water molecules were also calculated.Structural parameters comprising, for example, the —OH distance (e.g.,oxygen of the sulfonic acid/sulfonate group to the acidic hydrogen) andthe O . . . H . . . OH₂ distance (e.g., the distance from the oxygen onthe sulfonic acid/sulfonate group to the oxygen of the watermolecule/hydronium ion) are generally provided in Table 1 vide infra. Itis of some interest to note that despite the fact that CF₃SO₃H isgenerally regarded as a ‘superacid’, no substantial dissociation of theproton occurs with either the addition of the first water molecule oreven after a second water molecule is added. The CF₃SO₃H+H₂Oconformation generally demonstrates that the water molecule forms asomewhat shorter (as compared to the typical ˜2.8 Å) hydrogen bond withthe acidic proton and adopts an overall ‘six-membered ring’ conformationwith the SO₃H group. Table 1 demonstrates that the SO₃—H bond distancehas increased by about 0.086 Å after the second water molecule has beenadded over that observed in the minimum energy conformation of, forexample, CF₃SO₃H (0.973 Å). However, no dissociation of the proton isobserved even after a second water molecule is added.

After a third water molecule is added, a substantially spontaneousdissociation of the acidic proton may be observed during theB3LYP/6-31G** optimization. The formation of a hydronium ion is favoredthrough inter alia the formation of hydrogen bonds with the two watermolecules and one of the oxygens of the newly-formed triflate anion. Thedissociated state is generally adopted as a result of, for example, theexcess positive charge stabilized in the hydrogen bonding network andthe excess electron density due to the breaking of the SO₃—H bond whichis sufficiently delocalized by the electron withdrawing —CF₃ group. Thecombination of these two effects generally results in a minimum energyconformation for the cluster having a dissociated proton. Theseparation, as measured by the distance of the oxygen on the hydroniumion to the sulfonate oxygen from which the proton left, is on the orderof magnitude of approximately the mean of that observed by experiment.

The clusters formed with four and five water molecules are similar tothat observed with three water molecules in that the hydronium iongenerally forms a contact ion pair with the triflate anion. However, thehydronium ion typically adopts a position further away from the anion asthe number of water molecules is increase from 3 to 5. See, for example,Table 1.

Finally, with the addition of the sixth water molecule, a substantiallycomplete separation of the excess proton (e.g., hydronium ion) from theanion may be observed. This result was consistently demonstrated withoptimizations started from several different initial geometries. Here,the hydronium ion forms an Eigen ion as it is hydrogen bonded to threewater molecules with an average O—O distance of 2.56 Å. Of furthersignificance is the observation that the hydronium ion is nearly twicethe distance away from the anion (4.243 Å) as was observed in thecontact ion pair minimum energy conformations (2.556 Å-2.693 Å). Thissuggests that with sufficient water (i.e., with 6 H₂O's) the proton isshielded from direct electrostatic interaction with the sulfonate anion.Clearly, in the context of the hydrophilic terminations of a NAFIONpolymer, this observation will be considered to have significant effectson the conductivity of the membrane. This result also suggests that the‘first’ hydration shell of the triflate anion is made up of five watermolecules.

Optimized structures (at the B3LYP/6-31G** level) for para-toluenesulfonic acid with the successive addition of one through six watermolecules were also calculated. In comparing these minimum energyconformations with those obtained from triflic acid, a number ofqualitative similarities may be observed: (1) the conformation with asingle water molecule adopts the same ‘six-membered ring’ structure withthe SO₃H group; (2) upon the addition of the third water molecule, theproton spontaneously dissociates from the acid forming a hydronium ion;and (3) separation of the hydronium ion from the sulfonate anion doesnot occur until six water molecules are added. There are, however,important quantitative differences in the structural parameters of thevarious water clusters.

A similar set of structural parameters was tabulated for para-toluenesulfonic acid as generally given in Table 1. The oxygen-hydrogen bonddistance in the minimum energy conformation of both acids, without theaddition of any water molecules, is essentially the same (0.97 Å).However, after the addition of the first and second water molecules,this bond stretches to a greater extent in triflic acid (compare, forexample, columns 2 and 4 in Table 1) than with para-toluene sulfonicacid. Additionally, the water molecule hydrogen bonded to the acidicproton generally adopts a closer position to the sulfonic acid (compare,for example, columns 3 and 5 in Table 1) for the former system. Afterdissociation of the proton occurs (at n=3), the opposite trend isgenerally observed. Here, the hydronium ion in the water clusters of thearomatic sulfonic acid generally adopts a position that is closer to thesulfonate anion than that observed in the perfluorinated sulfonic acid.This trend continues with the further addition of water molecules; andafter separation of the ions occurs (at n=6), the difference in theseparation distances of the two ions is substantially greater withtriflic acid. All of these conformational differences may, at least in a‘Lewis acid’ sense, be rationalized in terms of the differences in thestrength of the acids and conjugate bases.

Triflic acid is generally a substantially stronger acid thanpara-toluene sulfonic acid. Upon dissociation of the first proton,electron density on the sulfonate anion is generally delocalized in bothsystems. However, in the aromatic system, electron density is typicallydelocalized in the π-ring; and in the perfluorinated system it iswithdrawn and stabilized by, for example, the —CF₃ group. The conjugateLewis base (i.e., sulfonate anion) that is formed is typically strongerin the case of para-toluene sulfonate anion than for the triflate anion.The strength of the conjugate base has a direct bearing on the positionof the dissociated proton; for example, the proton will interact morestrongly in the case of the stronger conjugate base, i.e. para-toluenesulfonate. This will be quantified as a function of the number of watermolecules in the acid/anion cluster through observation of the computedpartial atomic charges.

The atom centered partial charges and dipole moments (as calculated withthe ChelpG routine) of the water clusters of CF₃SO₃H and CH₃C₆H₄SO₃H arepresented in Tables 2 and 3. Examination of Table 2 shows thesubstantial positive charge residing on the sulfur atom and the negativecharge on the oxygen atoms. The charge on the former decreasessignificantly after dissociation, while the negative charge on thelatter increases upon dissociation of the proton. It is also interestingto note that there remains slightly more negative charge on the oxygenfrom which the proton resided throughout the addition of the watermolecules than on the other two sulfonate oxygens. TABLE 1 Structuralparameters for water clusters of CF₃SO₃H + nH₂O and CH₃C₆H₄SO₃H + nH₂O;(n = 1, 2, 3 . . . 6) CF₃SO₃H CF₃SO₃H CH₃C₆H₄SO₃H CH₃C₆H₄SO₃H Number ofWater r(—SO₃H . . . OH₂) r(—SO₃H . . . OH₂) r(—SO₃H . . . OH₂) r(—SO₃H .. . OH₂) Molecules, n Å Å Å Å 0 0.973 — 0.972 — 1 1.020 2.595 1.0072.650 2 1.059 2.496 1.033 2.564 3 1.562 2.556 1.437 2.488 4 1.721 2.6581.455 2.500 5 1.739 2.693 1.433 2.487 6 3.679 4.243 3.196 3.645

TABLE 2 Atom centered partial charges and dipole moment for CF₃SO₃H +nH₂O; (n = 1, 2, 3 . . . 6) Number of Total Water Atom, charge onMolecules, —SO₃H oxygen Dipole n S O^(†) O O atoms Moment 0 0.8436−0.4866 −0.4308 −0.3794 −1.2968 2.7219 1 0.7950 −0.5085 −0.4652 −0.3889−1.3626 3.3996 2 0.8460 −0.5178 −0.5074 −0.4106 −1.4358 4.2398 3 0.6484−0.5373 −0.4887 −0.4879 −1.5139 4.0042 4 0.6185 −0.5226 −0.5134 −0.4881−1.5241 1.8784 5 0.5812 −0.5163 −0.4887 −0.4533 −1.4583 2.6558 6 0.63.76−0.4978 −0.4703 −0.4703 −1.4384 4.8486O^(†) = oxygen atom to which acidic proton is/was bonded

TABLE 3 Atom centered partial charges and dipole moment forCH₃C₆H₄SO₃H + nH₂O; (n = 1, 2, 3 . . . 6) Number of Total Water Atom,charge on Molecules, —SO₃H oxygen Dipole n S O^(†) O O atoms Moment 00.9136 −0.5482 −0.4483 −0.4861 −1.4826 4.8514 1 0.7884 −0.5636 −0.4520−0.5041 −1.5197 2.6625 2 0.7354 −0.5803 −0.4884 −0.5048 −1.5735 3.4458 30.7459 −0.5947 −0.5512 −0.5511 −1.6970 2.0463 4 0.8359 −0.6048 −0.5995−0.5538 −1.7581 1.0546 5 0.8121 −0.6108 −0.5882 −0.5573 −1.7563 3.1181 60.6556 −0.5493 −0.5073 −0.5073 −1.6018 5.6153O^(†) = oxygen atom to which acidic proton is/was bonded

It is interesting to note the similarity in the negative excess chargedensity lying on the atom centers of the molecule throughout the rangeof associated water molecules; there is only slightly more negativecharge on CF₃SO₃H/CF₃SO₃ ⁻ than residing on CH₃C₆H₄SO₃H/CH₃C₆H₄SO₃ ⁻.Examination of the total negative charge residing on thesulfonic/sulfonate oxygen atoms (see, for example, the sixth column inTables 2 and 3) shows differences between the two systems. For example,there is consistently greater negative charge on the oxygens in thearomatic clusters. This demonstrates inter alia the increased basicityof the CH₃C₆H₄SO₃ ⁻ anion. The strength of the conjugate Lewis base foreither anion, therefore, substantially depends on the position adoptedby hydronium ion; i.e., a stronger conjugate base will result in acloser equilibrium position for the hydronium ion.

Incremental water binding energies were calculated for the addition ofthe six water molecules to both sulfonic acids according to therelation:ΔE _(b) =E[acid (H₂O)_(n) ]−E[H₂O ]−E[acid (H₂O)_(n-1)]

where total electronic energies are those computed at the B3LYP/6-31G**level of theory. The numerical values are presented in Table 4. Incomparing the two sulfonic acids, the binding energies generallydemonstrate that the water molecules in the clusters with triflic acidare somewhat more loosely bound to the acid than with para-toluenesulfonic acid. It may also be observed that with the aromatic acid thereis very little difference in the binding energies for the addition ofthe first 5 water molecules; the sixth water molecule is much moreloosely bound having a substantially lower binding energy. The latterwould suggest that the first hydration shell for p-toluene sulfonic acidprobably also comprises 5 water molecules. TABLE 4 Incremental bindingenergies (ΔBE) and standard free energies (ΔG°) (kcal/mol) for theaddition of the n ^(th) water molecule to CF₃SO₃H and CH₃C₆H₄SO₃H Numberof Water CF₃SO₃H CF₃SO₃H CH₃C₆H₄SO₃H CH₃C₆H₄SO₃H Molecules, n ΔBE ΔG°ΔBE ΔG° 1 −17.4 −5.6 −16.1 −3.7 2 −15.6 −5.3 −16.7 −7.0 3 −21.0 −6.1−16.4 −1.5 4 −16.6 −4.4 −16.0 −5.2 5 −20.3 −6.6 −16.2 −2.0 6 −19.2 −5.6−21.3 −7.2

The change in the standard Gibbs free energy for the reactions are alsotabulated in Table 4. These were calculated from G^(o)'s computed withG98 (Gaussian 98, Revision A.9; available from Gaussian Inc.,Pittsburgh, Pa., USA) employing the harmonic oscillator approximation.It is interesting to note that only relatively small fluctuations areobserved in the ΔG^(o)'s for triflic acid while the ΔG^(o)'s computedfor the toluene sulfonic acid show substantial variance. Comparing thesefree energy changes reveals that the dissociation of the proton (n=3 forboth acids) is much more favorable in the perfluorinated acid (˜−6.1 vs.˜−1.5 kcal/mol).

Friction and diffusion coefficients were computed for both NAFION 117and 65% sulfonated PEEKK membranes at ambient temperature (298.15K),each at three distinct hydration concentrations. The input parametersneeded in these computations were taken from small-angle X-rayscattering (SAXS) measurements. See, for example, K. D. Kreuer, J.Membr. Sci. 185, 29 (2001); M. Ise, Ph.D. Thesis, University ofStuttgart (2000); and T. D. Gierke et al., J. Polym. Sci. Polym. Phys.19, 1687 (1981). This information is presented in Table 5 and includes:the radius of the pore R, the length of the pore L, the total number ofwater molecules in the pore N, the total number of fixed sites in thepore ƒ_(s), the number of axially positioned radially symmetric arraysof sulfonate groups n, the average separation distance of the sulfonategroups d_(so) ₃ ⁻ , the average radial distance the hydronium ion isfrom the sulfonate groups r, and the amplitude of the periodic potentialΨ_(o). The numerical value of the last parameter is generally based onthe assumption that the dominant contribution to the proton mobility istypically due to the hydronium ion moving along the center of the pore.TABLE 5 Transport model input parameters for NAFION and 65% sulfonatedPEEKK membrane pores 65% 65% 65% Para- NAFION NAFION NAFION sulfonatedsulfonated sulfonated meter: 117 117 117 PEEKK PEEKK PEEKK λ 6 13 22.515 23 30 R, Å 8 14 16 7 9.5 12 L, Å 30 56 64 40 48 56 N 216 1001 1800375 828 1470 f_(s) 36 77 80 25 36 49 n 6 8 8 5 6 7 d_(SO) ₃ ⁻, Å 6 7 8 99 9 r, Å 4 10 12 6.4 9 11 ψ_(o), J 5.17E−22 2.10E−23 4.67E−24 3.63E−223.69E−23 7.48E−24

Examination of the parameters in Table 5 generally demonstrates that atsimilar degrees of hydration, the NAFION membrane pores comprisesubstantially larger diameters. It also may be observed that withincreasing membrane hydration, the size of the water-ion domains (e.g.,pores) increases beyond that due solely to the increase in the number ofwater molecules associated with each fixed site. This trend is generallydemonstrated for the perfluorinated and aromatic membranes and istracked by observing the increase in the value of the total number offixed sides ƒ_(s) and the total number of water molecules N. With thearomatic membranes, the average separation distance of the sulfonategroups remains relatively constant (e.g., ˜9 Å) over the range ofhydration. This, however, is generally not observed for the NAFIONmembranes, where with increasing hydration there is typically anincrease in the separation of the fixed sites. It is also notable thatthe amplitude of the electrostatic field due to the presence of theanionic groups along the walls of the pore ranges over nearly two ordersof magnitude. Accordingly, it may be appreciated that the increase inthe radius of the pore has a substantial impact on the electrostaticfield experienced by a hydronium ion moving along the center of thepore.

Friction coefficient correction terms (ζ₂, ζ₃, ζ₄) were computed atequally spaced intervals of 1 Å along the length of each pore. Averagevalues of these correction terms, along with the selected ‘base value’of the friction coefficient ζ_(i) were then used to calculate the protondiffusion coefficient.

The choice of the ‘base value’ of the friction coefficient ζ₁ isgenerally based on consideration of the permittivity of the water in thepore. The numerical value of ζ₁ was taken to be either that computedwith the Stokes relation for a hydronium ion in bulk water (˜2.69 E-12kg/s), or that derived from experimental diffusion measurements of aproton in bulk water (˜4.42 E-13 kg/s). See, for example, David R. Lide,Ed., Handbook of Chemistry and Physics, 80^(th) Ed. 5-94, CRC Press,Boca Raton (1999). For membrane hydration levels where the water in thepores is relatively bound through hydrophilic/electrostatic interactionswith the —SO₃ ⁻groups, the proton is transported as H3O⁺ suggesting theappropriateness of the former value of ζ₁. For membranes at higher waterconcentrations, where the water in the pores is more like bulk water,transport of the proton occurs (at least to some extent) throughtransfer from water molecule to water molecule (i.e., the Grotthussmechanism); and therefore, the latter value for ζ₁ is used as theapplicable value.

A model using an equilibrium statistical mechanical formulation wasdeveloped to compute the permittivity of water in PEM pores as afunction of: (1) pore size; (2) distribution of the fixed sites; and (3)radial separation distance of the proton from the anionic groups. See,for example, R. Paul and S. J. Paddison, J. Chem. Phys. 115, 7762(2001). The permittivity of water in the pores of NAFION membranes atwater concentrations of λ=6, 13, 22.5 suggest that much of the water atthe two lower water concentrations is bound through interaction with the—SO₃ ⁻ fixed sites. This is in contrast to the results computed forfully hydrated NAFION where a significant portion of the water issimilar to bulk water. With these results, the value for ζ₁ at the twolower hydration states was taken to be that derived with the Stokesrelation and for that at the highest water concentration, the valuederived from experimental proton diffusion measurements in bulk water.

Computed proton diffusion coefficient D_(α) for NAFION membranes arecompared with experimentally measure values D_(exp) in the plotrepresentatively depicted, for example, in FIG. 7. For experimentalvalues see, for example, K. D. Kreuer, J. Membr. Sci. 185, 29 (2001); M.Ise, Ph.D. Thesis, University of Stuttgart (2000); and T. A. Zawodzinskiat al., Electrochim. Acta 40, 297 (1995). This comparison reveals thatin all cases the calculated values are slightly lower (specifically,from about 8% to about 15% lower) than the experimental values.Accordingly, the agreement with experiment is quite good—on the order ofwithin the error of the measurements. While no contribution fromintermolecular (water-water) proton transfer was included in thecalculation of proton diffusion coefficients for the NAFION membranepores (at λ=6 and 13), this was included for the fully hydrated membrane(i.e., λ·222.5) in the choice of ζ₁.

A similar criteria and procedure were used to select a value for ζ₁ forPEEKK membranes at each hydration concentration. The results indicatedthat the relative permittivity of the water in the center of a pore atthe lowest water concentration (i.e., λ=15) is on the order of about67—approximately 16% less than that of bulk water; while at the twohigher water concentrations, the permittivity of the water is about 80,even for distances of up to 1.5 Å from the center of the pore. Based onthese results, ζ₁ was taken to be the value computed with the Stokesrelation for a hydronium ion in bulk water (˜2.69 E-12 kg/s) for λ=15,and that derived from experimental diffusion measurements of a proton inbulk water (·4.42 E-13 kg/s) for λ=23 and 30.

The computed proton diffusion coefficients for the 65% sulfonated PEEKKmembranes are also representatively plotted in FIG. 6 along with thecorresponding experimentally measured values. See, for example, K. D.Kreuer, J. Membr. Sci. 185, 29 (2001); and M. Ise, Ph.D. Thesis,University of Stuttgart (2000). Again, agreement with pulsed fieldgradient NMR measurements is good with the calculated values slightlylower than the experimental values.

This consistent result of the calculated values being smaller than theexperimental values suggests that the exemplary embodiment disclosedvide supra may over-estimate the effect of the —SO₃ ⁻ groups inretarding the mobility of the proton. The electrostatic field generatedby the anionic fixed sites is perhaps too high due to the neglect of thepresence of the additional protons. Accordingly, these protons willgenerally increase the shielding over that due from the water moleculesof the hydronium ion from interaction with SO₃ ⁻ groups and reduce theeffects of the latter on the water molecules in the pore. Thus,inclusion of additional protons in alternative exemplary embodiments inaccordance with the instant invention would tend to provide a decreasein the magnitudes of the computed friction coefficient correction termsand a consequent increase in the calculated proton diffusioncoefficient. The main reason the presence of additional protons were notincluded in the representative transport model in accordance with theexemplary embodiment disclose vide supra was due inter alia to thedistribution of these protons not having been characterized to arelatively degree of certainty. While the prior art has assumed aBoltzmann distribution for the protons within the pore (see, forexample, M. Eikerling et al., J. Phys. Chem. B 105, 3646 (2001); and M.Eikerling et al., J. Phys. Chem. 101, 10807 (1997)), an exemplaryembodiment of the present invention demonstrates that the prior artapproach generally neglects proton dissociation effects due todifferences in conjugate anionic bases. Moreover, such a continuumdistribution would tend to over-estimate shielding effects in pores withradii of less than two Debye lengths. Nevertheless, inclusion ofadditional protons in the model would be considered to provide improvedresults in alternative exemplary embodiments, and those skilled in theart will appreciate that the same are within the scope and ambit of thepresent invention.

Differential Sidechain Chemistry

Di-trifluoromethane ether was selected as a canonical representation forthe ether sidechain linkage of NAFION and a first candidate structure.Ab Initio calculations at the B3LYP/6-31G** level of theory wereundertaken for the fluoro-ether candidate moiety in vacuo as well aswith one water of hydration. As generally depicted, for example, in FIG.5, the O—O distance between the fluoro-ether oxygen 500 and the oxygenof the water 510 was determined to be 3.2 Å. A second candidate moietycorresponding to dimethyl ether was selected and optimized at the samelevel of theory, both in vacuo and with one water of hydration. Asgenerally depicted, for example, in FIG. 6, the O—O distance between thedimethyl ether oxygen 600 and the oxygen of the water 610 was 2.8 Å.Accordingly, the dimethyl ether system was considered to present ahigher hydrophilicity relative to the fluoro-ether candidate moiety(e.g., the canonical representation for the NAFION sidechain). Thisconsequently motivated the alteration of the NAFION sidechain to yieldthe novel ionomeric composition of matter as generally depicted, forexample, in FIG. 4; wherein the —CF₂ groups geminal to the ethersidechain linkage of NAFION are substituted with —CH₂ groups.

Statistical mechanical characterization of proton transport, asdescribed vide supra, for NAFION yielded the relative permittivityvalues of water as a function of hydration order and radial distancefrom the pore wall, as generally depicted, for example, in FIG. 8; aswell as the parametric sensitivity of proton diffusion as a function ofthe length of sidechain pore intrusion, as generally depicted, forexample, in FIG. 9. Similar characterization for the novel ionomericcomposition of matter representatively depicted in FIG. 4 demonstratedincreased sidechain pore intrusion with a substantial increase in thetendency of the ether oxygen to form a hydrogen bond with water.Additionally, modeling of the permittivity of the water in the pores ofthe novel ionomeric material in accordance with one exemplary embodimentof the present invention, demonstrated that the character of the waterwas altered so as to inhibit the transport of the water by the protoniccurrent, thereby preventing or otherwise ameliorating theelectro-osmotic drag from increasing. Additionally, since thehydrophilicity of the sidechains in alternative exemplary embodimentsmay be tailored so as to make the terminal moiety more hydrophilic thanalong the length of the sidechain, the conductivity of the membrane atlower water concentrations will typically not be reduced with increasinghydrophilicity of the sidechain.

The present invention may be described herein in terms of variousprocessing steps. It should be appreciated that such processing stepsmay be realized by any number of hardware and/or software componentsconfigured to perform the specified functions. For example, the presentinvention may employ various integrated circuit components, e.g., memoryelements, processing elements, logic elements, matchable datastructures, and the like, which may carry out a variety of functionsunder the control of one or more microprocessors or other controldevices.

Similarly, the software elements of the present invention may beimplemented with any programming or scripting language such as, forexample, Fortran, HPFortran, C, C++, Java, COBOL, assembler, PERL,extensible Markup Language (XML), etc., or any programming or scriptinglanguage now known or hereafter derived in the art, with the variousalgorithms being implemented with any combination of data structures,objects, processes, routines or other programming elements. Further, itshould be noted that the present invention may employ any number ofconventional techniques for data transmission, signaling, dataprocessing, parallelization, distributed processing, network control,and the like. Still further, the invention may employ various securitymeasures to prevent or otherwise deter, for example, code de-compilationwith inter alia client-side scripting languages, such as JavaScript,VBScript and/or the like. Alternatively, conjunctively or sequentially,the present invention may also employ cryptographic features designed toprotect access to data files and/or de-compilation of executable code.For a basic introduction of cryptography, please review, for example,the text written by Bruce Schneider entitled “Applied Cryptography:Protocols, Algorithms, And Source Code In C,” published by John Wiley &Sons (second edition, 1996).

It should be appreciated that the particular implementations shown anddescribed herein are illustrative of the invention and its best mode andare not intended to otherwise limit the scope of the present inventionin any way. Indeed, for the sake of brevity, conventional dataprocessing, application development and other functional aspects of thesystems (and components of the individual operating components of thesystems) may not be described in detail herein. Furthermore, dataprocessing components of various embodiments in accordance with thepresent invention are intended to provide exemplary functionalrelationships and/or couplings between the various elements. It shouldbe noted that many alternative or additional functional relationships orphysical connections may be present in a practical system.

It will be appreciated, that many applications of the present inventionmay be formulated. One skilled in the art will appreciate, for example,that a distributed processing architecture may include any system forexchanging data, such as, for example, the Internet, an intranet, anextranet, WAN, LAN, satellite communications, and/or the like. It isnoted that a network may be implemented as other types of networks, suchas an interactive television (ITV) network as well. The users mayinteract with the system via any input device such as a keyboard, mouse,kiosk, personal digital assistant, handheld computer (e.g., PalmPilot®),cellular phone and/or the like. Similarly, the invention could be usedin conjunction with any type of personal computer, network computer,workstation, minicomputer, mainframe, or the like running any operatingsystem such as any version of Windows, Windows XP, Windows ME, WindowsNT, Windows2000, Windows 98, Windows 95, MacOS, OS/2, BeOS, Linux, UNIX,or any operating system now known or hereafter derived by those skilledin the art. Moreover, the invention may be readily implemented withTCP/IP communications protocols, IPX, Appletalk, IP-6, NetBIOS, OSI orany number of existing or future protocols. Moreover, the systemcontemplates the use, sale and/or distribution of any goods, services orinformation having similar functionality described herein.

The computing units may be connected with each other via a datacommunication network. The network may be a public network and assumedto be insecure and open to eavesdroppers. In one exemplaryimplementation, the network may be embodied as the Internet. In thiscontext, the computers may or may not be connected to the Internet atall times. Specific information related to data traffic protocols,standards, and application software utilized in connection with theInternet may be obtained, for example, from Dilip Naik, InternetStandards and Protocols (1998); Java2 Complete, various authors, (Sybex1999); Deborah Ray and Eric Ray, Mastering HTML 4.0 (1997). Loshin,TCP/IP Clearly Explained (1997). A variety of conventionalcommunications media and protocols may be used for data links, such as,for example, a connection to an Internet Service Provider (ISP), overthe local loop as is typically used in connection with standard modemcommunication, cable modem, Dish networks, ISDN, Digital Subscriber Line(DSL), or various wireless communication methods. Data processingsystems in accordance with the present invention might also residewithin a local area network (LAN) which interfaces to a network via aleased line (T1, T3, etc.). Such communication methods are well known inthe art, and are covered in a variety of standard texts. See, e.g.,Gilbert Held, Understanding Data Communications (1996), herebyincorporated by reference.

As will be appreciated by one of ordinary skill in the art, the presentinvention may be embodied as a method, a system, a device, and/or acomputer program product. Accordingly, the present invention may takethe form of an entirely software embodiment, an entirely hardwareembodiment, or an embodiment combining aspects of both software andhardware. Furthermore, the present invention may take the form of acomputer program product on a computer-readable storage medium havingcomputer-readable program code means embodied in the storage medium. Anysuitable computer-readable storage medium may be utilized, includinghard disks, CD-ROM, optical storage devices, magnetic storage devices,and/or the like.

Data communication may be accomplished through any suitablecommunication means, such as, for example, a telephone network,intranet, Internet, point of interaction device (personal digitalassistant, cellular phone, kiosk, etc.), online communications, off-linecommunications, wireless communications, and/or the like. One skilled inthe art will also appreciate that, for security reasons, any databases,systems, or components of the present invention may consist of anycombination of databases or components at a single location or atmultiple locations, wherein each database or system includes any ofvarious suitable security features, such as firewalls, access codes,encryption, decryption, compression, decompression, and/or the like.

Where the instant invention embodies a method for performing the varioustasks disclosed herein as a software embodiment, computer programinstructions may be loaded onto a general purpose computer, specialpurpose computer, or other programmable data processing apparatus toproduce a machine, such that the instructions which execute on thecomputer or other programmable data processing apparatus create meansfor implementing the functions specified in the disclosed methods. Thesecomputer program instructions may also be stored in a computer-readablememory capable of directing a computer or other programmable dataprocessing apparatus to function in a particular manner, such that theinstructions stored in the computer-readable memory produce an articleof manufacture including instruction means which implement the functionsspecified in the disclosed method steps. The computer programinstructions may also be loaded onto a computer or other programmabledata processing apparatus to cause a series of operational steps to beperformed on the computer or other programmable apparatus to produce acomputer-implemented process such that the instructions which execute onthe computer or other programmable apparatus provide steps forimplementing the functions specified in the variously disclosed methodsteps.

Accordingly, the disclosed method steps support combinations of meansfor performing the specified functions, combinations of steps forperforming the specified functions, and program instruction means forperforming the specified functions. It will also be understood that eachdisclosed method step and combinations of method steps may beimplemented by either special purpose hardware-based computer systemswhich perform the specified functions or steps, or suitable combinationsof special purpose hardware and computer instructions.

In the foregoing specification, the invention has been described withreference to specific exemplary embodiments; however, it will beappreciated that various modifications and changes may be made withoutdeparting from the scope of the present invention as set forth in theclaims below. The specification and figures are to be regarded in anillustrative manner, rather than a restrictive one and all suchmodifications are intended to be included within the scope of thepresent invention. Accordingly, the scope of the invention should bedetermined by the claims appended hereto and their legal equivalentsrather than by merely the examples described above. For example, thesteps recited in any method or process claims may be executed in anyorder and are not limited to the specific order presented in the claims.Additionally, the components and/or elements recited in any apparatusclaims may be assembled or otherwise operationally configured in avariety of permutations to produce substantially the same result as thepresent invention and are accordingly not limited to the specificconfiguration recited in the claims.

Benefits, other advantages and solutions to problems have been describedabove with regard to particular embodiments; however, any benefit,advantage, solution to problems or any element that may cause anyparticular benefit, advantage or solution to occur or to become morepronounced are not to be construed as critical, required or essentialfeatures or components of any or all the claims.

As used herein, the terms “comprises”, “comprising”, or any variationthereof, are intended to reference a non-exclusive inclusion, such thata process, method, article, composition or apparatus that comprises alist of elements does not include only those elements recited, but mayalso include other elements not expressly listed or inherent to suchprocess, method, article, composition or apparatus. Other combinationsand/or modifications of the above-described structures, arrangements,applications, proportions, elements, materials or components used in thepractice of the present invention, in addition to those not specificallyrecited, may be varied or otherwise particularly adapted by thoseskilled in the art to specific environments, manufacturingspecifications, design parameters or other operating requirementswithout departing from the general principles of the same.

1. An ionomeric composition of matter suitably adapted for use as apolymer electrolyte membrane, said composition of matter comprising themonomeric chemical formula: CF₃(CH₂)_(m)O(CH₂)_(n)CF₂X ; wherein m is aninteger between 1 and 3, n is an integer between 1 and 3, and Xrepresents at least one of: an at least partially hydrophilic functionalgroup and an at least partially acidic functional group.
 2. Theionomeric composition of matter of claim 1, comprising at least one ofthe monomeric chemical formulas: CF₃CH₂OCH₂CF₂SO₃ and CF₃CH₂OCH₂CF₂SO₃H.3. A method for reducing the electro-osmotic drag of an at leastpartially hydrated polymer electrolyte membrane, said method comprisingthe steps of providing a chemical structure for a first ionomer andchemically modifying the sidechain of said first ionomer to produce asecond ionomer wherein the hydrophilicity of the sidechain of saidsecond ionomer is at least greater than the hydrophilicity of thesidechain of said first ionomer.
 4. The method for reducing theelectro-osmotic drag of an at least partially hydrated polymerelectrolyte membrane of claim 3, wherein said first ionomer comprises asulfonic acid based ionomer.
 5. The method for reducing theelectro-osmotic drag of an at least partially hydrated polymerelectrolyte membrane of claim 4, wherein said first ionomer comprises atleast one of NAFION and PEEKK.
 6. The method for reducing theelectro-osmotic drag of an at least partially hydrated polymerelectrolyte membrane of claim 3, wherein said chemical modification ofthe sidechain of said first ionomer comprises at least one of insertionand substitution of at least one ether oxygen along the length of thesidechain.
 7. The method for reducing the electro-osmotic drag of an atleast partially hydrated polymer electrolyte membrane of claim 3,wherein: said chemical modification of the sidechain of said firstionomer comprises substitution of at least a first atom having a firstelectronegativity with at least a second atom having a secondelectronegativity; said second electronegativity at least less than saidfirst electronegativity.
 8. The method for reducing the electro-osmoticdrag of an at least partially hydrated polymer electrolyte membrane ofclaim 7, wherein said first atom is fluorine (F) and said second atom ishydrogen (H).
 9. The method for reducing the electro-osmotic drag of anat least partially hydrated polymer electrolyte membrane of claim 8,wherein said chemical modification of a functional moiety comprising thechemical formula —O(CF₂)_(n)— results in a functional moiety comprisingthe formula —O(CH₂)_(n)— where n is an integer between 1 and
 3. 10. Themethod for reducing the electro-osmotic drag of an at least partiallyhydrated polymer electrolyte membrane of claim 8, wherein said chemicalmodification of a functional moiety comprising the chemical formula—CF₂OCF₂— results in a functional moiety comprising the formula—CH₂OCH₂—. 11-20. (canceled)